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Quantitative Illumination of Convex Bodies and Vertex Degrees of Geometric Steiner Minimal Trees
Author(s) -
Swanepoel Konrad J.
Publication year - 2005
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300000322
Subject(s) - mathematics , combinatorics , vertex (graph theory) , unit sphere , regular polygon , steiner tree problem , normed vector space , ball (mathematics) , unit (ring theory) , upper and lower bounds , discrete mathematics , geometry , mathematical analysis , graph , mathematics education
Two results are proved involving the quantitative illumination parameter B ( d ) of the unit ball of a d ‐dimensional normed space introduced by Bezdek (1992). The first is that B ( d ) = O (2 d d 2 log d ). The second involves Steiner minimal trees. Let v ( d ) be the maximum degree of a vertex, and s ( d ) that of a Steiner point, in a Steiner minimal tree in a d ‐dimensional normed space, where both maxima are over all norms. Morgan (1992) conjectured that s ( d ) ≤ 2 d , and Cieslik (1990) conjectured that v ( d ) ≤ 2(2 d − 1). It is proved that s ( d ) ≤ v ( d ) ≤ B ( d ) which, combined with the above estimate of B ( d ), improves the previously best known upper bound v ( d ) < 3 d .